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Cauchy sequences and Cauchy completions

Analysis

The notion of a Cauchy sequence goes back to work of Bolzano and Cauchy; it
provides a criterion for convergence. The construction of the real numbers from
the rationals via equivalence classes of Cauchy sequences is due to Cantor
[Ca1872] and Méray [Me1869]. In
fact, Charles Méray was apparently the first to provide a rigorous theory of
irrational numbers, shortly before Georg Cantor. The achievement of Méray is
well explained in a biographical note by Abraham Robinson
[Ro2008]. About Méray's article "Remarques sur la nature
des quantités définies par la condition de servir de limites à des variables
données" from 1869 he writes: "The paper marked the first appearance in print of
an 'arithmetical' theory of irrational numbers. Some years earlier Weierstrass
had, in his lectures, introduced the real numbers as sums of sequences or, more
precisely, indexed sets, of rational numbers; but he had not published his
theory and there is no trace of any influence of Weierstrass' thinking on
Méray's. Dedekind also seems to have developed his theory of irrationals at an
earlier date, but he did not publish it until after the appearance of Cantor's
relevant paper in 1872."

The construction of the real numbers via equivalence classes of Cauchy sequences
admits an obvious generalisation in the setting of metric spaces, leading to the
completion of a metric space.

Categories

For categories there is a notion of Cauchy completion that was introduced in a
paper by Borceux and Dejean [BD1986]. For a given category
$\cal C$, this identifies with the idempotent completion, which one obtains by taking
the category of all retracts of representable functors in the category of
functors from $\cal C$ to the category of sets. A more sophisticated
notion of Cauchy completion for enriched categories is due to Lawvere
[La1973]; it connects to the completion of metric
spaces when a metric space is viewed as an enriched category.

Cauchy sequences: The notion of a Cauchy sequence can be defined in any category $\cal C$ as a sequence
of morphisms $X_0 \to X_1 \to X_2 \to \cdots$ satisfying the following: for every object $C$
there exists $N_C\ge 0$ such that for all $n \ge m \ge N_C$ the
morphism $X_m \to X_n$ induces a
bijection $\Hom(C,X_m) \to \Hom(C,X_n)$ [Kr2018]. This
leads to another notion of Cauchy completion of $\cal C$.

Completion via Cauchy sequences:
For a category $\cal C$ there is a natural construction of a completion
$\hat{\cal C}$ that contains $\cal C$ as a full subcategory so
that each Cauchy sequence in $\cal C$ admits a colimit in $\hat{\cal C}$.
A morphism between sequences $(X_n)$ and $(Y_n)$ is given by a compatible sequence of morphisms
$(X_n \to Y_n)$ in $\cal C$, and one calls this eventually invertible if for any
object $C$ the induced map $\Hom(C,X_n) \to \Hom(C,Y_n)$ is bijective for
$n \gg 0$. The completion of $\cal C$ is now obtained from the category of
all Cauchy sequences (with the obvious morphisms) by formally inverting the
eventually invertible morphisms (so by attaching formal inverses). This yields a
new category $\hat {\cal C}$, and $\cal C$ identifies with the full subcategory of
constant sequences. The morphisms $X \to Y$ in $\hat {\cal C}$ can be identified
with equivalence classes of pairs of morphisms $X \to Y' \leftarrow Y$ such that
$Y \to Y'$ is eventually invertible, by taking such a pair to the composition of
$X \to Y'$ with the inverse of $Y \to Y'$.

Examples

Example 1: View the set $\mathbb{Q}$ of rational numbers with the usual ordering
as a category (with a unique morphism $x \to y$ if and only if $x \leq y$). Then
the completion of $\mathbb{Q}$ identifies with the ordered set $\mathbb{R}$ of
real numbers (plus an element $\infty$), by taking a Cauchy sequence to its
limit, if the sequence is bounded, and to $\infty$ otherwise.

Example 2: Fix any ring and consider the category $\cal C$ of modules of finite composition
length. Under a mild finiteness condition on $\cal C$, the
completion of $\cal C$ identifies with the category of all artinian
modules, by taking a Cauchy sequence to its colimit.

Example 3: Fix a right coherent ring and consider the category
$\cal C$ of perfect complexes. The full subcategory of the
completion of $\cal C$ given by the objects having bounded
cohomology identifies with the bounded derived category of
finitely presented modules, by taking a Cauchy sequence to its colimit.

Further examples will be added here as they become available. Everybody is
welcome to submit such examples (by sending a message to Henning Krause).